𝑉 = ℝ2
; 𝐻 ≔ {(𝑥, 𝑦) ∈ ℝ2
: 𝑥 = 𝑦}

Let's break down the given information:

1. **\( V = \mathbb{R}^2 \)**: This notation indicates that \( V \) is the vector space consisting of all ordered pairs of real numbers. So, any element in \( V \) has the form \( (x, y) \) where \( x, y \in \mathbb{R} \).

2. **\( H \equiv \{(x, y) \in \mathbb{R}^2 : x = y\} \)**: This defines a subset \( H \) of \( V \), where each element in \( H \) is an ordered pair \( (x, y) \) such that the first and second components are equal, i.e., \( x = y \).

### Interpretation:

- \( V = \mathbb{R}^2 \) is the entire 2-dimensional plane.
- \( H \) represents a line in this plane where every point on the line has equal \( x \) and \( y \) coordinates. This line is often called the **diagonal line** or the **line \( y = x \)**.

#### Key Points:
- \( H \) is a **subspace** of \( V \) since it satisfies the conditions for being a subspace (it's non-empty, closed under addition, and closed under scalar multiplication).
- Geometrically, \( H \) is a line passing through the origin at a 45-degree angle to both the \( x \)-axis and the \( y \)-axis.

Would you like to explore the properties of this subspace further, or do you have any specific questions about this topic?

### Related Questions:
1. What is a basis for the subspace \( H \)?
2. How do you determine the dimension of the subspace \( H \)?
3. Can \( H \) be considered as an eigenvector space? If so, for what matrix?
4. What is the orthogonal complement of \( H \) in \( V \)?
5. How would you project a vector in \( V \) onto \( H \)?
6. How does \( H \) relate to other common subspaces like \( \{(x, 0) \} \) or \( \{(0

Explore the relationship between vector spaces and subspaces with a focus on V = ℝ^2 and H = {(x, y) ∈ ℝ^2 : x = y}. Learn how H forms a subspace of V, its geometric interpretation, and properties. Suitable for undergraduate students studying linear algebra and vector spaces.

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